Unique continuation of the normal operator of the x-ray transform and applications in geophysics
Abstract
We show that the normal operator of the x-ray transform in ${\mathbb{R}}^{d}$?-->Rd , d ≥ 2, has a unique continuation property in the class of compactly supported distributions. This immediately implies uniqueness for the x-ray tomography problem with partial data and generalizes some earlier results to higher dimensions. Our proof also gives a unique continuation property for certain Riesz potentials in the space of rapidly decreasing distributions. We present applications to local and global seismology. These include linearized travel time tomography with half-local data and global tomography based on shear wave splitting in a weakly anisotropic elastic medium.
- Publication:
-
Inverse Problems
- Pub Date:
- April 2020
- DOI:
- 10.1088/1361-6420/ab6e75
- arXiv:
- arXiv:1909.05585
- Bibcode:
- 2020InvPr..36d5014I
- Keywords:
-
- inverse problems;
- x-ray tomography;
- normal operator;
- unique continuation;
- theoretical seismology;
- Mathematics - Functional Analysis;
- Mathematics - Classical Analysis and ODEs;
- Physics - Geophysics
- E-Print:
- 21 pages, 4 figures